Why are "batsmen" becoming "batters"?

Presumably you're referring to cricket, where 'batsman' is still by far the standard; 'batter' is only slightly increasing in usage recently. My guess about the reasons for the rise of 'batter':

1. The influence from American English, specifically baseball: 'batter' is the term in baseball.

2. With the rise of women's cricket (for some loose definition of 'rise') and the feeling in some quarters that 'batsman' is inappropriate for women.

'Batter' actually has a history of use in cricket. A 1934 book, The language of cricket, says of 'batter' that it was "Formerly used concurrently with 'batsman', as an alternative term." Similarly, M. A. Pervez's A Dictionary of Cricket (2001) defines 'batter' as:

A nineteenth-century term for a batsman; now used mainly in women's cricket to mean 'batswoman'.

The point is that as late as 2001, 'batter' was considered mainly an archaic term.

I found a discussion of 'batter' in this 2005 book:

‘Women's cricket’ is virtually ignored by the mainstream media […] Some would argue that the imperatives of changing cultural mores require the ‘neutralization’ of cricket's lexicon so that, for example, batsman becomes ‘batter’. Others would argue that because of the linguistic and cultural literary traditions of cricket, everyone should become a ‘batsman’. The first narrative wants to tell, or at least begin, a new story about cricket, creating a new text and a new textual practice. The second view wants women to be treated as true cricketers according to the historically validated practices of the game. […] It would appear for the time being that the latter view has won out in England. As Rachel Heyhoe Flint explains:

Another move firmly to establish women as cricketers was taken at the International Women's Cricket Council in Melbourne in 1985 'that all players should be known as batsman—not batswomen or batspersons’.

Seems a mess; we'll see how this plays out in a few years.

I have seen "batter" around, but 'batsman' is still more common. Looking at Cricinfo, the commentary to yesterday's England v Ireland game (perhaps you don't want to be reminded? :p) uses only 'batsman', not 'batter'. Elsewhere there is 'batter' used in quotes, and it seems to use both 'batswoman' and 'batter' for women's cricket (but it seems to me they more often find ways to avoid using either term!). (Only nine results for 'batswoman' in the last ten years.) So it's inconsistent, and again, probably the dust will settle in a few years and we'll arrive at consensus on terminology.